This article presents a general approximation-theoretic framework to analyze measure-transport algorithms for sampling and characterizing probability measures. Sampling is a task that frequently arises in data science and uncertainty quantification. We provide error estimates in the continuum limit, i.e., when the measures (or their densities) are given, but when the transport map is discretized or approximated using a finite-dimensional function space. Our analysis relies on the regularity theory of transport maps, as well as on classical approximation theory for high-dimensional functions. A third element of our analysis, which is of independent interest, is the development of new stability estimates that relate the normed distance between two maps to the divergence between the pushforward measures they define. We further present a series of applications where quantitative convergence rates are obtained for practical problems using Wasserstein metrics, maximum mean discrepancy, and Kullback-Leibler divergence. Specialized rates for approximations of the popular triangular Kn{\"o}the-Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory.

翻译：本文提出了一种通用的逼近理论框架，用于分析概率测度的采样与表征中的度量传输算法。采样是数据科学与不确定性量化中经常出现的任务。我们提供了连续极限下的误差估计，即当测度（或其密度）已知，但传输映射通过有限维函数空间进行离散或逼近时的情形。我们的分析依赖于传输映射的正则性理论以及高维函数的经典逼近理论。分析中的第三个要素（此要素具有独立研究价值）是建立了新的稳定性估计，该估计将两个映射之间的赋范距离与它们定义的推前测度之间的散度联系起来。我们进一步展示了系列应用实例，其中利用Wasserstein度量、最大均值差异和Kullback-Leibler散度获得了实际问题中的定量收敛速率。此外，我们针对流行的三角Könöthe-Rosenblatt映射的逼近给出了专门速率，并通过数值实验验证并拓展了我们的理论。